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C₂×C₁₃ #78

order	26
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×C₄ (order 12) · Inn(G) = C₁ · Out(G) = C₃×C₄
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₆
abelian	C₂×C₁₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 26

A minimal faithful permutation representation — 1 generator on 15 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15)

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₁₃, Q₈×C₁₃, Dic₃×C₁₃, …
derived subgroup of	Dic₂₆, D₅₂, C₂²⋊D₁₃, …
Frattini subgroup of	Dic₁₆₉, C₄×C₁₆₉, #13062557, …
Fitting subgroup of	Dic₁₃, C₂×D₁₃, C₂×C₁₃⋊C₃, …
socle of	Dic₁₃, C₄×C₁₃, C₂×D₁₃, …
maximal subgroup of	Dic₁₃, C₄×C₁₃, C₂×D₁₃, …
maximal normal subgroup of	Dic₁₃, C₄×C₁₃, C₂×D₁₃, …
maximal quotient of	C₄×C₁₃, C₂²×C₁₃, D₃×C₁₃, …
outer automorphism group of	D₅₃, C₇₉⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26, 2);